Structure of minimum-error quantum state discrimination
We consider minimum-error quantum state discrimination, and show its general properties useful to derive solutions in the minimum-error discrimination. We show that, for a set of quantum states to discriminate among, there exists a single positive operator, which we call a symmetry operator, that completely characterizes optimal parameters in minimum-error state discrimination, such as optimal measurement and minimal errors. Using symmetry operators, we introduce equivalent classes among sets of quantum states: those sets in the same class are defined as they share an identical symmetry operator, meaning that their optimal discrimination is already characterized. In the other way around, from a given symmetry operator, we provide a systematic way of constructing a set of quantum states for which optimal discrimination can be found by the operator. We then provide a geometric formulation for solving minimum-error state discrimination, exploiting the underlying geometry of given quantum states. All these obtained general properties are then illustrated with qubit states and solutions to various cases of multiple qubit states are obtained. We explicitly show how to find optimal measurement and a symmetry operator for a given set of qubit states. On the fundamental aspects, we first show that, when ensemble steering is applied with shared entanglement, steerability of quantum states in the ensemble is proportional to the success probability in state discrimination under the no-signaling condition. We also show equivalent expressions of the success probability, according to different approaches of solving state discrimination. Finally, we observe a distinguished feature of an asymptotic quantum cloning as state discrimination: relations of given quantum states conditionally play a role in quantum cloning, depending on whether quantum cloning is non-asymptotic or asymptotic.